DIFFUSION AND ENTROPY OF GASES 603 



state of the system, and cannot change while that state remains 

 unchanged. Therefore, if the system be a gas, at all events if a 

 perfect gas, it may be expressed as a function of the tem- 

 perature and volume — and, it may be, also a function of the 

 number, n, of molecules in the given volume, say generally it is 

 f (n, T, v). 



(d) But, if the gas be the working substance in a Carnot 

 cycle, the heat drawn from the reservoir in an isothermal 

 expansion from volume v to volume v + Sv is proportional to 

 the pressure, that is, for given v, proportional to n. Therefore, 

 also, the heat SQ, which passes into the refrigerator in the same 

 cycle, is proportional to n. Therefore, if S n be the entropy of a 

 single gas containing n molecules in volume v, S n < the entropy 



for the same gas containing n' molecules in volume v, ^^ = — , 



And since S n and S n < cannot differ by a constant, ~- n = -,. That 



o n ' n 



is, !•/"' V V x = " f° r all values of n and n'. This requires 

 ' f(n,T,v) n H 



that f (n, T, v) = n$ (T, v), where </> (T, v) denotes some function 



of T and v. Therefore f (n + n', T, v) = (n + n') <£ (T, v) 



= n<£ (T, v) + ricf) (T, v), or the entropy of a single gas, in 



volume v, is the sum of the entropies of all the parts into which 



it may be conceived as divided, provided that each part is free 



to move throughout the whole of v, and is not confined by 



partition to a part of v. (N.B. — This is my own argument, and 



not Bryan's. It would not necessarily apply to two different 



gases, because the physical state of each might be affected by 



the presence of the other.) Bryan, however, proves the same 



thing for two gases as follows : 



(e) The entropy of a gas occupying volume v is the same 

 whether it has that volume all to itself, or shares it with another 

 gas at the same temperature and pressure. In other words 

 each gas is in respect of its entropy a vacuum to the other. 

 Query, is there any limit to this law as the density increases ? 



A proof of this is given at p. 124 by the following method, 

 which I understand to be founded on experiment. Given a 

 mixture of two gases in cylinder of volume 2v, each having the 

 same partial pressure hp, and the mixture having combined 

 pressure p, the mixture is cooled until the gases liquefy, one 

 before the other. The liquids are put in separate vessels, 



